three-dimensional fluid-structure interaction analysis for ...once the csd model is made, the cfd...
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Abstract—In bridge design practice, especially in case of large
span bridges, wind loading can be extremely dangerous. Since
the collapse of the Tacoma-Narrows Bridge in 1941, bridge
flutter assessment has become a major concern in bridge design.
In the early ages, wind tunnel tests were made in order to assess
the aerodynamic performance of bridges. These tests required
scaled models of the bridges representing the structure by
insuring certain similarity laws. The wind tunnel models can be
either full-aeroelastic models or section models. The full models
are more detailed and precise while the section models can show
a two-dimensional slice of the bridge deck only. Such wind
tunnel tests are really expensive and time consuming tools in
bridge design therefore there is a strong demand to replace
them. In this paper a novel approach for bridge deck flutter
assessment based on numerical simulation will be presented.
Index Terms—bridge aeroelasticity, fluid-structure
interaction, three-dimensional simulation
I. INTRODUCTION
Nowadays, with a strong computational background, CFD
(Computational Fluid Dynamics) simulations appear to be
powerful rivals of the wind tunnel tests. Recently a number of
numerical simulations have been made aiming at determining
the aerodynamic performance of an ordinary bridge deck
section. These simulations are two-dimensional mainly for the
sake of time effectiveness [1]-[4]. The main shortcoming of
the two-dimensional approach is that it cannot capture the
rather complex three-dimensional coupling of the bridge deck
motion and the fluid flow around it (see Fig. 1).
There are case when a two-dimensional study can give
fairly good results for instance when the flow around a certain
long section of the bridge deck can be regarded as
two-dimensional. This is the case when a completed bridge is
studied; the flow around the bridge deck is nearly
two-dimensional, which means that the flow patterns around
different sections are similar. However, if a construction stage
is considered, the three-dimensional flow pattern must be
taken into account. Such case can be seen in Fig. 2. In this
picture the construction of a cable stayed bridge in South
Korea is shown. Construction stages are really challenging
both from structural dynamics and fluid dynamics point of
view.
Manuscript received August 16, 2009. The CFD.hu Ltd., and the
Pont-Terv Ltd supported this work.
G. Szabó is with the University of Technology and Economics Budapest,
Hungary (corresponding author to provide phone: 0036-30-327-9262; fax:
0036-1-2262096; e-mail: hoeses@freemail.hu).
J. Györgyi is with the University of Technology and Economics
Budapest, Hungary (gyorgyi@ep.mech.me.bme.hu).
Fig. 1: Tacoma Narrows Bridge flutter (sketch).
At the end of the cantilever (see Fig. 2) the flow field is
strongly three-dimensional, which makes the problem really
complicated. In these cases the two-dimensional approach
might provide unrealistic results. For bridge flutter prediction
it is hard to find three-dimensional coupled simulations in
literature, but for wing flutter analysis good results can be
found in [5]. The main goal in this study is to scrutinize the
very complicated three-dimensional response of a bridge deck
due to wind loading by using advanced numerical simulation
so that assumptions and simplifications should not be needed
during the modeling process.
For our purposes an efficient tool was required to handle
both the structural behavior and the fluid flow. The target
software has to offer coupling abilities between structural
dynamics and fluid dynamics. Considering the lack of
three-dimensional coupled numerical simulation in literature,
a wind tunnel model is to be made in order to have a chance
for validation.
Fig. 2: construction of the Incheon bridge in South Korea.
Three-dimensional Fluid-Structure Interaction
Analysis for Bridge Aeroelasticity
G. Szabó, J. Györgyi
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II. PROBLEM DISCRIPTION
Experiences in wind tunnel testing according to [6] and
two-dimensional CFD runs [7] were strongly required to start
a three-dimensional coupled numerical simulation. The
ANSYS 11.0 version was chosen with CSD (Computational
Solid Dynamics) and CFD (Computational Fluid Dynamics)
tools. By using the MFX multi-field solver involved in the
ANSYS, these two physics can be combined in one
simulation. As there are measurements for section models
only available, a fictive wind tunnel model was considered
which is now being constructed. This model is expected to
provide enough results to validate the MFX solver of the
ANSYS. This wind tunnel model is a two meters long full
aeroelastic model, which does not represent a real full-scale
bridge, but is really similar to usual wind tunnel models. The
bridge is a suspension bridge. The cables are modeled with
steel wires, which support the aluminium core beam. On the
core beam, balsa segments are fixed, which represent the
shape of the bridge. The cross section of our model can be
seen in Fig. 3.
Fig. 3: main dimensions of the wind tunnel model.
During the measurement the above detailed model will be
mounted in the wind tunnel. The inflow wind velocity is to be
varied within a certain range. At a certain wind speed the
model is expected to vibrate. The vibration amplitudes should
be measured. The wind speed, at which the model starts to
oscillate, is the critical wind speed of the flutter instability.
The critical wind speed is the target result from the study.
III. COMPUTATIONAL SOLID DYNAMICS
The wind tunnel model and its CSD (Computational Solid
Dynamics) model have to provide the same dynamic behavior
though it is not easy to accomplish. While the wind tunnel
model consist of a aluminium core beam bounded with a light
material which gives the shape of the bridge, the CSD model
is made up by using shell elements, by means of FEM (Finite
Element Method). The material properties and the thickness
value of the shell elements have to be adjusted in case of the
CSD model in order to reach similarity. In the fluid-structure
interaction calculation the force-displacement transfer on the
boundary of the structure and the fluid flow happens on the
bridge deck surface but the cable elements have a role on the
structural behavior only. In Fig. 4 the details of the FEM
model can be seen. The FEM mesh consists of regular
rectangular 4-node shell elements with six degrees of freedom
per each node while the cable elements are modeled with link
elements without bending capabilities. The surface of the
shell must be stiffened with diaphragms at every single cable
joints to the bridge deck.
Fig. 4: FEM model details of the bridge.
In the coupled simulation the deformation of the structure
is calculated at time steps. The ANSYS uses the Newmark
time advancement scheme. In this method the discrete
formulation of the structure is required (1), where M, C, and K
are the mass, damping, and stiffness matrices respectively
while x is the unknown displacement vector, q is the external
load vector. The discrete form of (1) can be seen in (2). The
solution methodology is as follows; at every single time step
the (2) linear equation system is solved for the x displacement
vector in the i+1 time step with the ∆t time step size. The
extended stiffness matrix and load vector are evaluated in (3)
and (4). In (5) and (6) the acceleration and velocity vectors
are determined respectively. The solver is unconditionally
stable by applying α and β stability parameters. In the
simulation the Rayleigh damping was used.
)()()()( tqtxKtxCtxM =++ &&&& (1)
11ˆ
++ = ii pxK (2)
Ct
Mt
KK∆
+∆
+=β
α
β 21ˆ (3)
[ ]
∆
−+
−+
∆+
−+∆+
∆+= ++
iii
iiiii
xtxxt
C
xtxxt
Mqp
&&&
&&&
12
1
12
11211
β
α
β
α
β
α
ββ (4)
[ ] iiiii xtxxxt
x &&&&&
−−∆−−
∆= ++ 1
2
11121 ββ
(5)
[ ] xtxxxt
x iiii &&&& ∆
−+
−+−
∆= ++
β
α
β
α
β
α
21111
(6)
-
Before applying the Newmark method, it is highly
recommended to perform a dynamic eigenvalue calculation.
The results from this simulation are the natural frequencies of
the system and the corresponding mode shapes. These
dynamic properties are really important to be understood; the
behavior of the system due to various load cases can be
predicted by studying them.
The first four mode shapes of the bridge model can be seen
in Fig. 5. The first and the second modes are asymmetrical and
symmetrical shapes respectively with pure vertical motion.
The third shape belongs to the torsion of the bridge deck. The
fourth one shows a horizontal motion of the bridge deck. The
natural frequencies are 1.61Hz, 2.45Hz, 6.33Hz, and, 9.55Hz
respectively. In the flutter phenomena the torsion mode has
the most important role, but according to recent flutter
studies, about the first ten shapes have to be included in the
simulation.
Fig. 5: the first four mode shapes of the FEM model.
IV. COMPUTATIONAL FLUID DYNAMICS
Once the CSD model is made, the CFD model is to be built.
For the CFD calculations the CFX module of the ANSYS
system was used. The software uses the Finite Volume
Method for the spatial discretization. This technique requires
subdividing the domain of the flow into cells. For the CFD
calculations a relatively coarse mesh was made to reduce the
computational efforts. The number of cells was around
200.000. The CFD mesh can be seen in Fig. 6. The numerical
mesh was made around the bridge contour in two-dimension
first. The meshing is finer around the bridge and coarser in the
farther reaches of the computational domain. To ensure that,
unstructured mesh was used in the plane perpendicular to the
bridge axis. This mesh was extruded along the bridge axis
providing a structured longitudinal mesh. The computational
domain can be seen in Fig. 7. In the vicinity of the bridge
contour the velocity and pressure gradients are high, however,
considering that the present study is preliminary, the coarse
mesh around the bridge contour is acceptable. Naturally
precise pressure distribution cannot be expected by this way.
Nevertheless, for the CSD calculation rough pressure
distribution calculation is enough in this stage of the research,
and probably the results are good in order.
Fig. 6: CFD mesh for the flow simulation.
-
Fig. 7: computational domain.
From the CFD simulation the pressure distribution is
needed. To obtain that, the Navier-Stokes equation (7) with
the continuity equation should be solved.
∂
∂+
∂
∂+
∂
∂+
∂
∂−=
∂
∂+
∂
∂+
∂
∂+
∂
∂2
x
2
2
x
2
2
x
2
xx
zx
yx
xx
zyxv
xρ
1g
zyxt
vvvpvv
vv
vv
v
∂
∂+
∂
∂+
∂
∂+
∂
∂−=
∂
∂+
∂
∂+
∂
∂+
∂
∂2
y
2
2
y
2
2
y
2
y
y
z
y
y
y
x
y
zyxv
yρ
1g
zyxt
vvvpvv
vv
vv
v
∂
∂+
∂
∂+
∂
∂+
∂
∂−=
∂
∂+
∂
∂+
∂
∂+
∂
∂2
z
2
2
z
2
2
z
2
z
z
z
z
y
z
x
z
z
v
y
v
x
vv
zρ
1g
zyxt
pvv
vv
vv
v
0zyx
zyx =∂
∂+
∂
∂+
∂
∂ vvv (7)
Firstly, the N-S equation is averaged in time, which gives
the so-called Reynolds stress tensor. This gives six more
unknowns therefore closure (turbulence) models have to be
applied. In CFD codes there are many turbulence models
available but unfortunately none of them is perfect for all kind
of flow patterns. In this work the k-ε model was used for its robustness and reliability. The k is the turbulent kinetic
energy and ε is the energy dissipation. In CFD applications this one is the most widely used.
V. FLUID-STRUCTURE INTERACTION
For the fluid-structure interaction simulation the CSD and
the CFD models are necessary. The geometry of them must be
conform. The key momentum in the coupled simulation is that
the fluid flow induces forces on the bridge deck, which
deforms accordingly by using the CSD calculations. The
deformations are fed back to the CFD mesh. The time step
was set to 0.0008 seconds. The end time was 0.48s, which
needed 6 days run time on a four core 2.40 GHz computer
with 4.0 GB RAM memories.
The inflow velocity was varied from 5 to 10m/s. From a
non-deformed initial state of the bridge, the development of
the flutter motion needs much time. Thus, the bridge deck was
loaded with a torsion moment for a certain transient time and
then it was released, and the free vibration was investigated
afterwards. If the amplitudes do not grow, the system could be
regarded as stable under these conditions. At the velocities of
5.0 and 7.5 m/s flutter does not occur. At the speed of 10.0 m/s
the vibration amplitudes grew from the initial value. This state
showed a typical flutter phenomenon.
VI. RESULTS
In Fig. 8 the flow around the deformed bridge deck is
shown at the speed of 7.5m/s at different time steps. As the
cable elements do not take part in the solid-fluid coupling,
they are not shown at all. In the total deformation the torsion
of the bridge deck dominates combined with a little vertical
displacement. The thin lines on the surface of the bridge show
the deformed CFD mesh.
The bridge shape is considered to be streamlined, so great
and intensive vortices are not expected to be present nor flow
separation from the body. This is why a coarse mesh might be
adequate for shapes like this.
In case of bridge sections with sharp edges (Fig. 1),
however, the flutter phenomenon is accompanied with strong
vortices. By using the k-ε or similar turbulence models the
accurate modeling of such flow patterns is unlikely. For
problems like that, LES (Large Eddy Simulation) turbulence
model is offered, which provides accurate results but at an
extremely high computational cost.
Fig. 8: streamlines around the deformed bridge deck.
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VII. CHECK OF THE RESULTS
Due to the rather complicated solution introduced above,
there is a need to check roughly its reliability. The dynamic
loading due to the airflow was strongly simplified for this
purpose; for different angles of attack the lift forces and
moments can be calculated assuming flat plate theory with the
following expressions:
).sin(2
22
γρ
π LBU
F = (8)
).sin(22
22
γρπ
LBU
M = (9)
In the expressions above ρ is the air density, U is the
airflow velocity, L is the section length, and B is the plate
width. The simplified aerodynamic forces as a function of the
angle of attack (γ) are shown in Fig. 9 for U=7.5m/s wind
speed. As it can be seen, in this angle of attack range, the
functions are almost linear.
The flow calculation time has been decreased radically by
using (8) and (9). To reduce that of the CSD model, it has
been simplified as well. This CSD model, made with the
ANSYS software, can be seen in Fig. 10. In this model the
cable elements were modeled by using simple beams with
tension-compression capabilities only while the aluminum
core beam and the balsa cross beams are modeled with beam
elements. The beam elements have six degrees of freedom per
each node. The bridge is fully constrained at both ends. The
beam model is able to approximate the behavior of that of the
previously introduced more complex model.
-1,20
-0,80
-0,40
0,00
0,40
0,80
1,20
-10 -8 -6 -4 -2 0 2 4 6 8 10
Gamma [degrees]
Lif
t fo
rce [
N];
Mo
men
t [N
m]
F
M
Fig. 9: aerodynamic forces with the flat plate theory.
Fig. 10: simplified beam model of the bridge.
Fig. 11: two mode shapes of the bridge model.
In Fig. 11 the mode shapes of the beam model can be seen.
Above the first mode represents a heave motion while below
the torsion mode is shown. Both shapes are very similar to the
corresponding modes of the complex CSD model presented
previously (see the first and the third shapes in Fig. 5).
A quasi-static approach can be applied by using (8) and (9).
To do so, the dynamic equation of motion of the bridge must
be written:
).()()()(0
tqtKxtxKtxMr
=++ &&&πω
δ (10)
In (10), M and K are the mass matrix and the stiffness
matrix respectively, q is the time dependent load vector, δ and
ω0r are the logarithmic decrement and the r-th circular
frequency of the bridge. The mode shape values of the beam
model (see Fig. 11) can be collected in a V vector. By using
the V vector and seeking the solution in the form of x=Vy,
(10) yields:
).()()()(0
tqVtKVyVtyKVVtyMVVTTT
r
T =++ &&&πω
δ (11)
.EMVV T = (12)
....... 202
0
2
01 nr
T KVV ωωω= (13)
).()()()()( 200 tftqvtytyty rT
rrrrrr ==++ ωωπ
δ&&& (14)
As it can be seen in (14) the differential equation of motion
of the bridge (10) is split into several parts so a number of
differential equation of motions have to be solved but
involving only one unknown function. By using this solution
technique, the frequency independency of the structural
damping can be modeled more precisely contrary to the
Rayleigh damping.
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In [8] detailed description can be found to solve (14). The
solution methodology is as follows; at every single time step
the angle of attack relative to the airflow must be determined
and for the next time step the lift force and the moment can be
calculated by using (8) and (9). For this calculation a program
was written with the Matlab software. This approach neglects
the induced flow forces due to the motion of the deck. To take
them into account, [9] proposes a solution for a
three-dimensional beam model including the aerodynamic
derivatives of an arbitrary bridge section.
To calculate the critical wind flutter speed of the bridge
model, the inflow velocity was increased step by step, and the
free damped oscillation of the structure was recorded as in the
case of the coupled CSD-CFD model.
In Fig. 12 the vertical motion of the middle point of the
bridge is shown at the wind speed of 5.0m/s with the
CSD-CFD and the simplified beam model. As it can be seen,
both models show a damped free oscillation after releasing
them.
As the CSD-CFD coupled model, the beam model was used
with increased wind velocities as well. At the speed of 10.0
m/s it also showed flutter-like condition. This means that for
this bridge model the critical wind speed of flutter can be
between 5.0 and 10.0 m/s. To find all the instabilities versus
the flow velocities, the CSD-CFD simulation should be done
at various flow velocities. The problem is that every run
requires much time. In addition, the present coarse CFD mesh
is not adequate for capturing the vortex shedding for instance.
Naturally, considering the results from the two strongly
different three-dimensional modeling techniques, perfect
coincidences cannot be expected. Nevertheless, the
agreement in order is very promising for the future work.
-0,004
-0,003
-0,002
-0,001
0,000
0,001
0,002
0,003
0,004
0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45 0,50
Time [s]
Dis
pla
cem
en
t [m
]
CSD-CFD
BEAM-FLATPLATE
Fig. 12: vertical displacement of the bridge
VIII. CONCLUSION
To conclude, in this paper a three-dimensional coupled
CSD-CFD simulation was performed. In an attempt to make a
validation in the near future, a fictive full aeroelastic wind
tunnel model was considered. The shell model of the bridge
was built up on the border of the fluid and the solid domain.
At the velocity inlet boundary in CFX, constant velocities
were defined, and the free vibrating motion of the bridge deck
was studied under airflow. The methodology for finding the
critical flutter speed was to apply an inlet velocity
increasingly in different runs. When the motion amplitude
started growing, the critical velocity was found.
At this stage of the research work the wind tunnel test has
not been finished, therefore some kind of approximation was
needed to check the results of the complex coupled model.
Thus, the critical wind speed was checked with a simplified
beam model of the bridge combined with a simple load model
of the wind. The guess value of the critical wind speed of the
two models was in the same order.
The presented three-dimensional CSD-CFD simulation has
strong difficulties, indeed, and cannot be regarded as a
solution for three-dimensional flutter prediction at this stage
of our research. Nevertheless, in the near future, the
aeroelastic wind tunnel model will be made so that the
methodology can be validated and become a reliable tool in
bridge aeroelasticity as well as in other civil engineering
applications.
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dynamic analyses of flutter characteristics for self-anchored
suspension bridges, Front. Archit. Civ. Eng. China 2008, 2(3):
267-273
[3] Theodorsen T., (1935): General theory of aerodynamic instability and
the mechanism of flutter. TR 496, NACA.
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around five generic bridge deck sections. J. Wind Engng. And
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[8] Bathe K. J., Wilson E. L. (1976): Numerical methods in finite element
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